Fading of transmitted signals can be a serious problem in cellular radio telephone systems. Intermittent loss of signal, even losing connections altogether, can result. Practical system capacity at any time is also important, and can be adversely affected at least temporarily by overall limit on transmit power and bandwidth. It is not sensible or practical to consider driving every available communications channel always at a power level sufficient to satisfy weakest possible transmission. Prior approaches to appropriate power control have included monitoring signals within a subscriber's channel relative to a predefined signal strength above which a base station will command reduction of transmitting power by subscriber equipment concerned. Unfortunately, such approach can actually contribute to signal degradation. Thus, if transmitting power is so reduced when it is wholly or mainly power of interference content that has increased, the signal (or carrier) to noise-plus-interference ratio (SNIR) will be made yet worse. Another approach has been to accept as inevitable that some connections will always be at risk, and, whether relative to indication of actual SNIR or otherwise, simply to seek to limit the channels at risk to a small percentage, say 10% or less, typically so that the majority, say 90% or more, are viable, i.e. at least effectively have SNIR above a given minimum value. Ordinarily, normal spread of actual SNIR values will mean that most channels are well above such given minimum value, and it is an object of this invention to provide power control with such channels closer to such given minimum value, so that at least some channels previously below SNIR threshold may attain an SNIR above threshold due to overall reduction in interference levels.